Finding the Area and Perimeter Distributions for Flat Poisson Processes of a Straight Line and Voronoi Diagrams

Abstract

The study of distribution functions (with respect to areas, perimeters) for partitioning a plane (space) by a random field of straight lines (hyperplanes) and for obtaining Voronoi diagrams is a classical problem in statistical geometry. Moments for such distributions have been investigated since 1972 [1]. We give a complete solution of these problems for the plane, as well as for Voronoi diagrams. The following problems are solved: 1. A random set of straight lines is given on the plane, all shifts are equiprobable, and the distribution law has the form \(F(\varphi ).\) What is the area (perimeter) distribution of the parts of the partition? 2. A random set of points is marked on the plane. Each point A is associated with a “region of attraction,” which is a set of points on the plane to which A is the closest of the marked set. The idea is to interpret a random polygon as the evolution of a segment on a moving one and construct kinetic equations. It is sufficient to take into account a limited number of parameters: the covered area (perimeter), the length of the segment, and the angles at its ends. We show how to reduce these equations to the Riccati equation using the Laplace transform.